Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C127 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16861150 |
| commutator | C16×C127 | C22 |
| Frattini | C16 | C2×D127 |
| Fitting | Q32×C127 | C2 |
| socle | C2×C127 | D16 |
| hypercenter | Q32 | D127 |
| hypocenter | C127 | Q64 |
| LCS | G ▷ C16×C127 ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
30 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,c^b=cd,d^b=de,e^b=ef,g^b=g^126712 conjugacy classes — view the full character table.
| maximal subgroup of | Q64×D127, C2×#22549716, #30641037, … |
| maximal normal subgroup of | Q64×D127, C2×#22549716, #30641037, … |
| maximal quotient of | #30636910, #30636911, #30636929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 389);